# Show that cos²π/10+cos²4π/10+cos² 6π/10+cos²9π/10=2. I am a bit confused if I make Cos²4π/10=cos²(π-6π/10) & cos²9π/10=cos²(π-π/10), it will turn negative as cos(180°-theta)=-costheta in the second quadrant. How do I go about proving the question?

Feb 21, 2018

#### Explanation:

$L H S = {\cos}^{2} \left(\frac{\pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right) + {\cos}^{2} \left(\frac{6 \pi}{10}\right) + {\cos}^{2} \left(\frac{9 \pi}{10}\right)$

$= {\cos}^{2} \left(\frac{\pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right) + {\cos}^{2} \left(\pi - \frac{4 \pi}{10}\right) + {\cos}^{2} \left(\pi - \frac{\pi}{10}\right)$

$= {\cos}^{2} \left(\frac{\pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right) + {\cos}^{2} \left(\frac{\pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right)$

$= 2 \cdot \left[{\cos}^{2} \left(\frac{\pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right)\right]$

$= 2 \cdot \left[{\cos}^{2} \left(\frac{\pi}{2} - \frac{4 \pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right)\right]$

$= 2 \cdot \left[{\sin}^{2} \left(\frac{4 \pi}{10}\right) + {\cos}^{2} \left(\frac{4 \pi}{10}\right)\right]$

$= 2 \cdot 1 = 2 = R H S$

Feb 21, 2018

We know that,
color(red)(costheta = sin (pi/2-theta) so also,
color(red)(cos^2theta = sin^2 (pi/2-theta)
color(magenta)(costheta = -sin((3pi)/2-theta) so also,
color(magenta)(cos^2theta = (-sin((3pi)/2-theta))^2 = sin^2((3pi)/2-theta)

getting back to the question,

color(red)(cos²π/10)+cos²(4π)/10+cos² (6π)/10+color(magenta)(cos²(9π)/10)=2

color(red)(sin²(pi/2-π/10))+cos²(4π)/10+cos² (6π)/10+color(magenta)((-sin((3pi)/2-(9π)/10))^2)=2

sin²((5pi)/10-π/10)+cos²(4π)/10+cos² (6π)/10+sin²((3pi)/2-(9π)/10)=2

[sin²(4π)/10+cos²(4π)/10]+[cos² (6π)/10+sin²((15pi)/10-(9π)/10)]=2

[sin²(4π)/10+cos²(4π)/10]+[cos² (6π)/10+sin²(6π)/10]=2

Applying, ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

$1 + 1 = 2$
$2 = 2$

Hence Proved.

P.S. you were going right, just note that even if its negative, the final answer turns out to be positive as the $\cos$ is squared according to the question. Any negative number squared is positive :)